While you can use R for this, it's easy enough to do from the command line:

sed 's/; /\t/g' foo.txt | awk '{for(i=2;i<NF;i+=1) print $1"\t"$i}' > foo.new.txt

Here is an R code example. There are probably better methods around (look at the tidyr and reshape2 packages for instance)

library(stringr)
library(foreach)
library(iterators)

test <- rbind(data.frame(miRNA="hsa-let-7a", genes="A;B;C;D"), data.frame(miRNA="hsa-mir-32", genes="D;E")
foreach(row=iter(test, "row"), .combine=rbind) %do% {
  genes <- str_split(row$genes, ";")
  foreach(gene=genes, .combine=rbind) %do% {
    return(data.frame(miRNA=row$miRNA, gene=gene))
  }
}

While not 100% directly related to your question, it may be nonetheless useful to you and others based on the topic title.

We have a free-to-use application called iPathwayGuide that will take gene-expression data and provide a list of potentially active miRNAs based on the gene-expression signature. We also display the target genes for each miRNA that have at least one DE target gene. See screen shot below.

In the "Genes" tab, you can see the miRNA's that target a specific gene.

We are working on developing a more generic "browser" that would allow you to see all the various inter-relationships between various entities such as genes, miRNAs, Ontologies, pathways, drugs, variants, and diseases from any perspective. Should be out sometime in Q2 of this year.